1 Fermi liquid behavior of the in-plane resistivity in the pseudogap state of YBa 2Cu 4O8 Cyril Proust 1*, B. Vignolle 1, J. Levallois 1,2, S. Adachi 3 and N. E. Hussey 4,5* 1 Laboratoire National des Champs Magnétiques Intenses (LNCMI-EMFL), (CNRS-INSA- UGA-UPS), Toulouse 31400, France. 2 Department of Quantum Matter Physics, University of Geneva, CH-1211 Geneva 4, Switzerland. 3 Superconductivity Research Laboratory, Shinonome 1-10-13, Koto-ku, Tokyo 135-0062, Japan. 4 High Field Magnet Laboratory (HFML-EMFL), Radboud University, Toernooiveld 7, 6525ED Nijmegen, Netherlands 5 Radboud University, Institute of Molecules and Materials, Heyendaalseweg 135, 6525 AJ Nijmegen, Netherlands * To whom correspondence may be addressed. Email: [email protected] or [email protected] . 2 Our knowledge of the ground state of underdoped hole-doped cuprates has evolved considerably over the last few years. There is now compelling evidence that inside the pseudogap phase, charge order breaks translational symmetry leading to a reconstructed Fermi surface made of small pockets. Quantum oscillations, (Doiron-Leyraud N, et al. (2007) Nature 447:564-568), optical conductivity (Mirzaei SI, et al. (2013) Proc Natl Acad Sci USA 110:5774-5778) and the validity of Wiedemann-Franz law (Grissonnache G, et al. (2016) Phys. Rev. B 93:064513) point to a Fermi liquid regime at low temperature in the underdoped regime. However, the observation of a quadratic temperature dependence in the electrical resistivity at low temperatures, the hallmark of a Fermi liquid regime, is still missing. Here, we report magnetoresistance measurements in the magnetic- 2 field-induced normal state of underdoped YBa 2Cu 4O8 which are consistent with a T resistivity extending down to 1.5 K. The magnitude of the T2 coefficient, however, is much smaller than expected for a single pocket of the mass and size observed in quantum oscillations, implying that the reconstructed Fermi surface must consist of at least one additional pocket. Significance High temperature superconductivity evolves out of a metallic state that undergoes profound changes as a function of carrier concentration, changes that are often obscured by the high upper critical fields. In the more disordered cuprate families, field suppression of superconductivity has uncovered an underlying ground state that exhibits unusual localization behavior. Here, we reveal that in stoichiometric YBa 2Cu 4O8, the field-induced ground state is both metallic and Fermi-liquid like. The manuscript also demonstrates the potential for using the absolute magnitude of the electrical resistivity to constrain the Fermi surface topology of correlated metals and in the case of YBa 2Cu 4O8, reveals that the current picture of the reconstructed Fermi surface in underdoped cuprates as a single, isotropic electron-like pocket may be incomplete. Introduction The generic phase diagram of Fig. 1 summarizes the temperature and doping dependence of the 1 in-plane resistivity ρab (T) of hole-doped cuprates . Starting from the heavily overdoped side, non-superconducting La 1.67Sr 0.33CuO 4 for example shows a purely quadratic resistivity below ~ 50 K (ref. 2). Below a critical doping pSC where superconductivity sets in, ρab (T) exhibits 2 n supra-linear behaviour that can be modeled either as ρab ~ T + T or as T (1 < n < 2). When a magnetic field is applied to suppress superconductivity on the overdoped side, the limiting low- T behavior is found to be T-linear 3,4,5. Optimally doped cuprates are characterized by a linear resistivity for all T > Tc, though the slope often extrapolates to a negative intercept suggesting 1 that at the lowest temperatures, ρab (T) contains a component with an exponent larger than one . In the underdoped regime, ρab (T) varies approximately linearly with temperature at high T, but as the temperature is lowered below the pseudogap temperature T*, it deviates from linearity in a very gradual way 6. At lower temperatures, marked by the light blue area in Fig. 1, there is now compelling evidence from various experimental probes of incipient charge order 7-11 . High 12,13 14 field NMR and ultrasonic measurements indicate that a phase transition occurs below Tc. This is also confirmed by recent high field x-ray measurements which indicate that the CDW order becomes tridimensional with a coherence length that increases with increasing magnetic field strength 15,16 . This leads to a Fermi surface reconstruction that can be reconciled with quantum oscillations 17,18 as well as with the sign change of the Hall 19 and Seebeck 20 coefficients. Whether the charge order is biaxial 21 or uniaxial with orthogonal domains 22 is still 3 an open issue, but a Fermi surface reconstruction involving two perpendicular wavevectors leads to at least one electron pocket in the nodal region of the Brillouin zone 23 . Depending on the initial pseudogapped Fermi surface and on the wavevectors of the charge order, Fermi surface reconstruction can also lead to additional, smaller hole pockets24,25 . In Y123 (doping level p ≈ 0.11) the quantum oscillation spectra consist of a main frequency Fa = 540 T and a beat pattern indicative of nearby frequencies Fa2 = 450 T and Fa3 = 630 T. A smaller frequency Fh ≈ 100 T has been detected by thermopower and c-axis transport measurements and attributed 26 there to an additional small hole pocket . The presence of the three nearby frequencies Fai can be explained by a model involving a bilayer system with an electron pocket in each plane and 27,28 magnetic breakdown between the two pockets . In this scenario, the low frequency Fh could originate from quantum interference or the Stark effect 29 . However, this scenario predicts the occurrence of five nearby frequencies, and thus requires fine tuning of certain microscopic parameters such as the bilayer tunneling t⊥. Moreover, the doping dependence of the Seebeck coefficient is difficult to reconcile with a Fermi surface reconstruction scenario leading to only one electron pocket per plane. More generally, the observation of quantum oscillations is a classic signature of Landau quasiparticles. In underdoped Y123, the temperature dependence of the amplitude of the oscillations follows Fermi-Dirac statistics up to 18 K, as in the Landau-Fermi liquid theory 30 . This conclusion is supported by other observations, such as the validity of the Wiedemann- Franz law 31 in underdoped Y123 and the quadratic frequency and temperature dependence of the quasiparticle lifetime τ (ω, T) measured by optical spectroscopy32 in underdoped HgBa 2CuO 4+ δ (Hg1201). An important outstanding question is whether the in-plane resistivity of underdoped cuprates also exhibits the behavior of a canonical Landau-Fermi liquid, namely a quadratic temperature dependence at low T? In underdoped cuprates, several studies have 2 shown ρab ~ T , but always at elevated temperatures (indeed, most are above Tc) and only over a limited temperature range that never exceed a factor of 2.5 6,33 -36 . (see Table S1 for a detailed 36 list of studies). At low temperatures, either ρab (T) starts to become non metallic , suggesting that the T 2 behavior observed at intermediate temperatures could just be a crossover regime, or a quadratic behaviour has previously been hinted at19 , rather than shown explicitly. Here, we present high field in-plane magnetoresistance measurements in underdoped YBa 2Cu 4O8 (Y124) 2 that are consistent with the form ρa(T) = ρ0(T) + AT from T = Tc down to temperatures as low as 1.5 K, e.g. over almost two decades in temperature. In addition, we investigate the magnitude of the resultant A coefficient and compare it with some of the prevalent Fermi surface reconstruction scenarios. In conclusion, we find that the magnitude of A is difficult to reconcile with the existence of a single electron pocket per plane, with an isotropic mass. Results We have measured the a-axis magnetoresistance (i.e. perpendicular to the conducting CuO chains) of two underdoped Y124 (Tc = 80 K) single crystals up to 60 T at various fixed temperatures down to 1.5 K. From thermal conductivity measurements at high fields, the upper 37 critical field Hc2 of Y124 has been estimated to be ~ 45 T . Raw data for both samples are shown in Fig. 2. Above Tc, i.e. in the absence of superconductivity, the transverse magnetoresistance can be accounted for, over the entire field range measured , by a two-carrier model 35 using the formula: (1) () = (0) + where ρ(0) is the zero-field resistivity and α and β are free parameters that depend on the conductivity and the Hall coefficient of the electron and hole carriers 35 (see Fig. S1 for a comparison of the two-band and single-band, quadratic forms for the magnetoresistance.) To 4 obtain reliable values of ρ (H→0, T) = ρ(0) and corresponding error bars for each field sweep, the data were fitted to equation (1) in varying field ranges using the procedure described in detail in Figs. S2-S5 and Tables S2-S3. Precisely the same form is used to fit the high-field data at all temperatures studied below Tc. This procedure has been found to yield reliable ρ(0) values in both cuprate 5 and pnictide 38 superconductors. Extrapolation of the high-field data to the zero- field axis ρ(0), as shown by dashed lines in Fig. 2, allows one then to follow the evolution of ρa(T) down to low temperatures. The extrapolated ρ(0) values are plotted versus temperature in Fig. 3 (symbols) for both crystals, along with the zero-field temperature dependence of the resistivity (solid line). The dashed lines in Fig. 3 correspond to fits of the ρ(0, T) data to the 2 n form ρ0(T) = ρ0 + AT .